Total-coloring of Sparse Graphs with Maximum Degree 6

Given a graph G = (V, E) and a positive integer k, a k-total-coloring of G is a mapping φ: V⋃E → {1, 2, ⋯, k} such that no two adjacent or incident elements receive the same color. The central problem of the total-colorings is the Total Coloring Conjecture, which asserts that every graph of maximum degree Δ admits a (Δ+2)-total-coloring. More precisely, this conjecture has been verified for Δ ≤ 5, and it is still open when Δ = 6, even for planar graphs. Let mad(G) denote the maximum average degree of the graph G. In this paper, we prove that every graph G with Δ(G) = 6 and mad(G) < 23/5 admits an 8-total-coloring.